You need to siphon water from a clogged sink. The sink has an area of 0.484 m2 and is filled to a height of h = 3.99 cm. Your siphon tube rises h1 = 40.6 cm above the bottom of the sink and then descends h2 = 103 cm to a pail as seen in the figure below.

The siphon tube has a diameter of 2.14 cm. Assuming that the water enters the siphon tube with almost zero velocity, calculate its velocity when it enters the pail.

To calculate the velocity of the water when it enters the pail, we can use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid in a steady flow.

Bernoulli's equation is given by:

P1 + 0.5 * ρ * v1^2 + ρ * g * h1 = P2 + 0.5 * ρ * v2^2 + ρ * g * h2

Here,
P1 and P2 are the pressures at points 1 and 2 respectively,
ρ is the density of the fluid,
v1 and v2 are the velocities of the fluid at points 1 and 2 respectively,
g is the acceleration due to gravity,
h1 and h2 are the heights of the fluid at points 1 and 2 respectively.

Let's calculate the velocities step by step:

1. Calculate the pressure difference between the two points:
ΔP = P1 - P2

2. Calculate the height difference between the two points:
Δh = h2 - h1

3. Rearrange Bernoulli's equation to solve for the velocity at point 2:
v2 = √(((P1 - P2) + ρ * g * (h1 - h2))/ρ)

Now, let's substitute the given values into the equation:

- The pressure difference ΔP can be taken as zero since we are assuming the water enters the siphon tube with almost zero velocity.

- The density of water is ρ = 1000 kg/m^3.

- The acceleration due to gravity is g = 9.8 m/s^2.

- The height difference Δh = h2 - h1 = 103 cm - 40.6 cm = 62.4 cm = 0.624 m.

Substituting these values into the equation:

v2 = √(((0) + (1000 kg/m^3) * (9.8 m/s^2) * (40.6 cm - 103 cm))/(1000 kg/m^3))

Simplifying the equation:

v2 = √((-5908.8)/1000)

v2 = √(-5.9088)

The velocity of the water when it enters the pail is imaginary since the term inside the square root is negative. This indicates that the water does not flow freely into the pail, and the siphon is not working properly.

In this case, you may need to check for any obstructions or issues with the siphon setup to ensure proper flow.